# Examples of Simplicial Groupoids in Nature

For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition.

Do you have examples simplicial groupoids that occur in nature? Here's what I have got:

1. Given a simplicial group $G$ acting on a simplicial set $X$, the action groupoid $X//G$ is a simplicial groupoid.
2. The fundamental groupoid $\Pi X$ of a bisimplicial set or simplicial space.
3. Given a functor $F:I\to {\mathbf{sSet}}$ from a diagram category $I$ to the category of simplicial sets, one can form what Goerss-Jardine calls the translation category $E_I F$ and what Mac Lane-Moerdijk calls the category of elements $\int_{I^{\mathrm{op}}}\, F$. The nerve of this category calculates the homotopy colimit ${\mathrm{hocolim}}_I \, F$. In the case where the diagram category is a groupoid, then this translation category/category of element is a simplicial groupoid.
4. Given a simplicial set $X$, the loop groupoid $GX$ is a simplicial groupoid.
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See some papers following

Ehlers, P.J. and Porter, T. Varieties of simplicial groupoids. I. Crossed complexes. J. Pure Appl. Algebra 120~(3) (1997) 221--233.